<s>
Bootstrapping	B-Application
is	O
any	O
test	O
or	O
metric	O
that	O
uses	O
random	O
sampling	O
with	O
replacement	O
(	O
e.g.	O
</s>
<s>
mimicking	O
the	O
sampling	O
process	O
)	O
,	O
and	O
falls	O
under	O
the	O
broader	O
class	O
of	O
resampling	B-General_Concept
methods	O
.	O
</s>
<s>
Bootstrapping	B-Application
assigns	O
measures	O
of	O
accuracy	O
(	O
bias	O
,	O
variance	O
,	O
confidence	O
intervals	O
,	O
prediction	O
error	O
,	O
etc	O
.	O
)	O
</s>
<s>
Bootstrapping	B-Application
estimates	O
the	O
properties	O
of	O
an	O
estimand	O
(	O
such	O
as	O
its	O
variance	O
)	O
by	O
measuring	O
those	O
properties	O
when	O
sampling	O
from	O
an	O
approximating	O
distribution	O
.	O
</s>
<s>
One	O
standard	O
choice	O
for	O
an	O
approximating	O
distribution	O
is	O
the	O
empirical	B-General_Concept
distribution	I-General_Concept
function	I-General_Concept
of	O
the	O
observed	O
data	O
.	O
</s>
<s>
In	O
the	O
case	O
where	O
a	O
set	O
of	O
observations	O
can	O
be	O
assumed	O
to	O
be	O
from	O
an	O
independent	O
and	O
identically	O
distributed	O
population	O
,	O
this	O
can	O
be	O
implemented	O
by	O
constructing	O
a	O
number	O
of	O
resamples	B-General_Concept
with	O
replacement	O
,	O
of	O
the	O
observed	O
data	B-General_Concept
set	I-General_Concept
(	O
and	O
of	O
equal	O
size	O
to	O
the	O
observed	O
data	B-General_Concept
set	I-General_Concept
)	O
.	O
</s>
<s>
It	O
is	O
often	O
used	O
as	O
an	O
alternative	O
to	O
statistical	O
inference	O
based	O
on	O
the	O
assumption	O
of	O
a	O
parametric	O
model	O
when	O
that	O
assumption	O
is	O
in	O
doubt	O
,	O
or	O
where	O
parametric	O
inference	O
is	O
impossible	O
or	O
requires	O
complicated	O
formulas	O
for	O
the	O
calculation	O
of	O
standard	B-General_Concept
errors	I-General_Concept
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
was	O
published	O
by	O
Bradley	O
Efron	O
in	O
"	O
Bootstrap	B-General_Concept
methods	I-General_Concept
:	O
another	O
look	O
at	O
the	O
jackknife	B-Algorithm
"	O
(	O
1979	O
)	O
,	O
inspired	O
by	O
earlier	O
work	O
on	O
the	O
jackknife	B-Algorithm
.	O
</s>
<s>
The	O
bias-corrected	O
and	O
accelerated	O
(	O
BCa	O
)	O
bootstrap	B-Application
was	O
developed	O
by	O
Efron	O
in	O
1987	O
,	O
and	O
the	O
ABC	O
procedure	O
in	O
1992	O
.	O
</s>
<s>
The	O
basic	O
idea	O
of	O
bootstrapping	B-Application
is	O
that	O
inference	O
about	O
a	O
population	O
from	O
sample	O
data	O
(	O
sample	O
→	O
population	O
)	O
can	O
be	O
modeled	O
by	O
resampling	B-General_Concept
the	O
sample	O
data	O
and	O
performing	O
inference	O
about	O
a	O
sample	O
from	O
resampled	O
data	O
(	O
resampled	O
→	O
sample	O
)	O
.	O
</s>
<s>
In	O
bootstrap-resamples	O
,	O
the	O
'	O
population	O
 '	O
is	O
in	O
fact	O
the	O
sample	O
,	O
and	O
this	O
is	O
known	O
;	O
hence	O
the	O
quality	O
of	O
inference	O
of	O
the	O
'	O
true	O
 '	O
sample	O
from	O
resampled	O
data	O
(	O
resampled	O
→	O
sample	O
)	O
is	O
measurable	O
.	O
</s>
<s>
More	O
formally	O
,	O
the	B-Application
bootstrap	I-Application
works	O
by	O
treating	O
inference	O
of	O
the	O
true	O
probability	O
distribution	O
J	O
,	O
given	O
the	O
original	O
data	O
,	O
as	O
being	O
analogous	O
to	O
an	O
inference	O
of	O
the	O
empirical	O
distribution	O
Ĵ	O
,	O
given	O
the	O
resampled	O
data	O
.	O
</s>
<s>
The	O
simplest	O
bootstrap	B-Application
method	I-Application
involves	O
taking	O
the	O
original	O
data	B-General_Concept
set	I-General_Concept
of	O
heights	O
,	O
and	O
,	O
using	O
a	O
computer	O
,	O
sampling	O
from	O
it	O
to	O
form	O
a	O
new	O
sample	O
(	O
called	O
a	O
'	O
resample	O
 '	O
or	O
bootstrap	B-Application
sample	I-Application
)	O
that	O
is	O
also	O
of	O
sizeN	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
sample	O
is	O
taken	O
from	O
the	O
original	O
by	O
using	O
sampling	O
with	O
replacement	O
(	O
e.g.	O
</s>
<s>
This	O
process	O
is	O
repeated	O
a	O
large	O
number	O
of	O
times	O
(	O
typically	O
1,000	O
or	O
10,000	O
times	O
)	O
,	O
and	O
for	O
each	O
of	O
these	O
bootstrap	B-Application
samples	I-Application
,	O
we	O
compute	O
its	O
mean	O
(	O
each	O
of	O
these	O
is	O
called	O
a	O
"	O
bootstrap	B-Application
estimate	O
"	O
)	O
.	O
</s>
<s>
We	O
now	O
can	O
create	O
a	O
histogram	O
of	O
bootstrap	B-Application
means	O
.	O
</s>
<s>
A	O
great	O
advantage	O
of	O
bootstrap	B-Application
is	O
its	O
simplicity	O
.	O
</s>
<s>
It	O
is	O
a	O
straightforward	O
way	O
to	O
derive	O
estimates	O
of	O
standard	B-General_Concept
errors	I-General_Concept
and	O
confidence	O
intervals	O
for	O
complex	O
estimators	O
of	O
the	O
distribution	O
,	O
such	O
as	O
percentile	O
points	O
,	O
proportions	O
,	O
odds	O
ratio	O
,	O
and	O
correlation	O
coefficients	O
.	O
</s>
<s>
However	O
,	O
despite	O
its	O
simplicity	O
,	O
bootstrapping	B-Application
can	O
be	O
applied	O
to	O
complex	O
sampling	O
designs	O
(	O
e.g.	O
</s>
<s>
for	O
population	O
divided	O
into	O
s	O
strata	O
with	O
ns	O
observations	O
per	O
strata	O
,	O
bootstrapping	B-Application
can	O
be	O
applied	O
for	O
each	O
stratum	O
)	O
.	O
</s>
<s>
Bootstrap	B-Application
is	O
also	O
an	O
appropriate	O
way	O
to	O
control	O
and	O
check	O
the	O
stability	O
of	O
the	O
results	O
.	O
</s>
<s>
Although	O
for	O
most	O
problems	O
it	O
is	O
impossible	O
to	O
know	O
the	O
true	O
confidence	O
interval	O
,	O
bootstrap	B-Application
is	O
asymptotically	O
more	O
accurate	O
than	O
the	O
standard	O
intervals	O
obtained	O
using	O
sample	O
variance	O
and	O
assumptions	O
of	O
normality	O
.	O
</s>
<s>
Bootstrapping	B-Application
is	O
also	O
a	O
convenient	O
method	O
that	O
avoids	O
the	O
cost	O
of	O
repeating	O
the	O
experiment	O
to	O
get	O
other	O
groups	O
of	O
sample	O
data	O
.	O
</s>
<s>
Bootstrapping	B-Application
depends	O
heavily	O
on	O
the	O
estimator	O
used	O
and	O
,	O
though	O
simple	O
,	O
naive	O
use	O
of	O
bootstrapping	B-Application
will	O
not	O
always	O
yield	O
asymptotically	O
valid	O
results	O
and	O
can	O
lead	O
to	O
inconsistency	O
.	O
</s>
<s>
Although	O
bootstrapping	B-Application
is	O
(	O
under	O
some	O
conditions	O
)	O
asymptotically	O
consistent	O
,	O
it	O
does	O
not	O
provide	O
general	O
finite-sample	O
guarantees	O
.	O
</s>
<s>
The	O
apparent	O
simplicity	O
may	O
conceal	O
the	O
fact	O
that	O
important	O
assumptions	O
are	O
being	O
made	O
when	O
undertaking	O
the	B-Application
bootstrap	I-Application
analysis	O
(	O
e.g.	O
</s>
<s>
Also	O
,	O
bootstrapping	B-Application
can	O
be	O
time-consuming	O
and	O
there	O
are	O
not	O
many	O
available	O
software	O
for	O
bootstrapping	B-Application
as	O
it	O
is	O
difficult	O
to	O
automate	O
using	O
traditional	O
statistical	O
computer	O
packages	O
.	O
</s>
<s>
Scholars	O
have	O
recommended	O
more	O
bootstrap	B-Application
samples	I-Application
as	O
available	O
computing	O
power	O
has	O
increased	O
.	O
</s>
<s>
Increasing	O
the	O
number	O
of	O
samples	O
cannot	O
increase	O
the	O
amount	O
of	O
information	O
in	O
the	O
original	O
data	O
;	O
it	O
can	O
only	O
reduce	O
the	O
effects	O
of	O
random	O
sampling	O
errors	O
which	O
can	O
arise	O
from	O
a	O
bootstrap	B-Application
procedure	O
itself	O
.	O
</s>
<s>
Moreover	O
,	O
there	O
is	O
evidence	O
that	O
numbers	O
of	O
samples	O
greater	O
than	O
100	O
lead	O
to	O
negligible	O
improvements	O
in	O
the	O
estimation	O
of	O
standard	B-General_Concept
errors	I-General_Concept
.	O
</s>
<s>
In	O
fact	O
,	O
according	O
to	O
the	O
original	O
developer	O
of	O
the	O
bootstrapping	B-Application
method	O
,	O
even	O
setting	O
the	O
number	O
of	O
samples	O
at	O
50	O
is	O
likely	O
to	O
lead	O
to	O
fairly	O
good	O
standard	B-General_Concept
error	I-General_Concept
estimates	O
.	O
</s>
<s>
recommend	O
the	B-Application
bootstrap	I-Application
procedure	O
for	O
the	O
following	O
situations	O
:	O
</s>
<s>
Since	O
the	O
bootstrapping	B-Application
procedure	O
is	O
distribution-independent	O
it	O
provides	O
an	O
indirect	O
method	O
to	O
assess	O
the	O
properties	O
of	O
the	O
distribution	O
underlying	O
the	O
sample	O
and	O
the	O
parameters	O
of	O
interest	O
that	O
are	O
derived	O
from	O
this	O
distribution	O
.	O
</s>
<s>
If	O
the	O
underlying	O
distribution	O
is	O
well-known	O
,	O
bootstrapping	B-Application
provides	O
a	O
way	O
to	O
account	O
for	O
the	O
distortions	O
caused	O
by	O
the	O
specific	O
sample	O
that	O
may	O
not	O
be	O
fully	O
representative	O
of	O
the	O
population	O
.	O
</s>
<s>
When	O
power	B-General_Concept
calculations	I-General_Concept
have	O
to	O
be	O
performed	O
,	O
and	O
a	O
small	O
pilot	O
sample	O
is	O
available	O
.	O
</s>
<s>
One	O
method	O
to	O
get	O
an	O
impression	O
of	O
the	O
variation	O
of	O
the	O
statistic	O
is	O
to	O
use	O
a	O
small	O
pilot	O
sample	O
and	O
perform	O
bootstrapping	B-Application
on	O
it	O
to	O
get	O
impression	O
of	O
the	O
variance	O
.	O
</s>
<s>
However	O
,	O
Athreya	O
has	O
shown	O
that	O
if	O
one	O
performs	O
a	O
naive	O
bootstrap	B-Application
on	O
the	O
sample	O
mean	O
when	O
the	O
underlying	O
population	O
lacks	O
a	O
finite	O
variance	O
(	O
for	O
example	O
,	O
a	O
power	O
law	O
distribution	O
)	O
,	O
then	O
the	B-Application
bootstrap	I-Application
distribution	O
will	O
not	O
converge	O
to	O
the	O
same	O
limit	O
as	O
the	O
sample	O
mean	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
confidence	O
intervals	O
on	O
the	O
basis	O
of	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
of	O
the	B-Application
bootstrap	I-Application
could	O
be	O
misleading	O
.	O
</s>
<s>
Athreya	O
states	O
that	O
"	O
Unless	O
one	O
is	O
reasonably	O
sure	O
that	O
the	O
underlying	O
distribution	O
is	O
not	O
heavy	O
tailed	O
,	O
one	O
should	O
hesitate	O
to	O
use	O
the	O
naive	O
bootstrap	B-Application
"	O
.	O
</s>
<s>
In	O
univariate	O
problems	O
,	O
it	O
is	O
usually	O
acceptable	O
to	O
resample	O
the	O
individual	O
observations	O
with	O
replacement	O
(	O
"	O
case	O
resampling	B-General_Concept
"	O
below	O
)	O
unlike	O
subsampling	O
,	O
in	O
which	O
resampling	B-General_Concept
is	O
without	O
replacement	O
and	O
is	O
valid	O
under	O
much	O
weaker	O
conditions	O
compared	O
to	O
the	B-Application
bootstrap	I-Application
.	O
</s>
<s>
In	O
small	O
samples	O
,	O
a	O
parametric	O
bootstrap	B-Application
approach	O
might	O
be	O
preferred	O
.	O
</s>
<s>
For	O
other	O
problems	O
,	O
a	O
smooth	O
bootstrap	B-Application
will	O
likely	O
be	O
preferred	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
is	O
generally	O
useful	O
for	O
estimating	O
the	O
distribution	O
of	O
a	O
statistic	O
(	O
e.g.	O
</s>
<s>
In	O
particular	O
,	O
the	B-Application
bootstrap	I-Application
is	O
useful	O
when	O
there	O
is	O
no	O
analytical	O
form	O
or	O
an	O
asymptotic	O
theory	O
(	O
e.g.	O
,	O
an	O
applicable	O
central	O
limit	O
theorem	O
)	O
to	O
help	O
estimate	O
the	O
distribution	O
of	O
the	O
statistics	O
of	O
interest	O
.	O
</s>
<s>
This	O
is	O
because	O
bootstrap	B-General_Concept
methods	I-General_Concept
can	O
apply	O
to	O
most	O
random	O
quantities	O
,	O
e.g.	O
,	O
the	O
ratio	O
of	O
variance	O
and	O
mean	O
.	O
</s>
<s>
There	O
are	O
at	O
least	O
two	O
ways	O
of	O
performing	O
case	O
resampling	B-General_Concept
.	O
</s>
<s>
The	O
Monte	O
Carlo	O
algorithm	O
for	O
case	O
resampling	B-General_Concept
is	O
quite	O
simple	O
.	O
</s>
<s>
First	O
,	O
we	O
resample	O
the	O
data	O
with	O
replacement	O
,	O
and	O
the	O
size	O
of	O
the	O
resample	O
must	O
be	O
equal	O
to	O
the	O
size	O
of	O
the	O
original	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
We	O
repeat	O
this	O
routine	O
many	O
times	O
to	O
get	O
a	O
more	O
precise	O
estimate	O
of	O
the	B-Application
Bootstrap	I-Application
distribution	O
of	O
the	O
statistic	O
.	O
</s>
<s>
The	O
'	O
exact	O
 '	O
version	O
for	O
case	O
resampling	B-General_Concept
is	O
similar	O
,	O
but	O
we	O
exhaustively	O
enumerate	O
every	O
possible	O
resample	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
This	O
can	O
be	O
computationally	O
expensive	O
as	O
there	O
are	O
a	O
total	O
of	O
different	O
resamples	B-General_Concept
,	O
where	O
n	O
is	O
the	O
size	O
of	O
the	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
Thus	O
for	O
n	O
=	O
5	O
,	O
10	O
,	O
20	O
,	O
30	O
there	O
are	O
126	O
,	O
92378	O
,	O
6.89	O
×1010	O
and	O
5.91	O
×1016	O
different	O
resamples	B-General_Concept
respectively	O
.	O
</s>
<s>
However	O
,	O
if	O
we	O
are	O
not	O
ready	O
to	O
make	O
such	O
a	O
justification	O
,	O
then	O
we	O
can	O
use	O
the	B-Application
bootstrap	I-Application
instead	O
.	O
</s>
<s>
Using	O
case	O
resampling	B-General_Concept
,	O
we	O
can	O
derive	O
the	O
distribution	O
of	O
.	O
</s>
<s>
We	O
first	O
resample	O
the	O
data	O
to	O
obtain	O
a	O
bootstrap	B-Application
resample	O
.	O
</s>
<s>
There	O
are	O
some	O
duplicates	O
since	O
a	O
bootstrap	B-Application
resample	O
comes	O
from	O
sampling	O
with	O
replacement	O
from	O
the	O
data	O
.	O
</s>
<s>
Also	O
the	O
number	O
of	O
data	O
points	O
in	O
a	O
bootstrap	B-Application
resample	O
is	O
equal	O
to	O
the	O
number	O
of	O
data	O
points	O
in	O
our	O
original	O
observations	O
.	O
</s>
<s>
Then	O
we	O
compute	O
the	O
mean	O
of	O
this	O
resample	O
and	O
obtain	O
the	O
first	O
bootstrap	B-Application
mean	O
:	O
μ1*	O
.	O
</s>
<s>
We	O
repeat	O
this	O
process	O
to	O
obtain	O
the	O
second	O
resample	O
X2*	O
and	O
compute	O
the	O
second	O
bootstrap	B-Application
mean	O
μ2*	O
.	O
</s>
<s>
This	O
represents	O
an	O
empirical	O
bootstrap	B-Application
distribution	O
of	O
sample	O
mean	O
.	O
</s>
<s>
From	O
this	O
empirical	O
distribution	O
,	O
one	O
can	O
derive	O
a	O
bootstrap	B-Application
confidence	O
interval	O
for	O
the	O
purpose	O
of	O
hypothesis	O
testing	O
.	O
</s>
<s>
In	O
regression	O
problems	O
,	O
case	O
resampling	B-General_Concept
refers	O
to	O
the	O
simple	O
scheme	O
of	O
resampling	B-General_Concept
individual	O
cases	O
–	O
often	O
rows	O
of	O
a	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
For	O
regression	O
problems	O
,	O
as	O
long	O
as	O
the	O
data	B-General_Concept
set	I-General_Concept
is	O
fairly	O
large	O
,	O
this	O
simple	O
scheme	O
is	O
often	O
acceptable	O
.	O
</s>
<s>
Therefore	O
,	O
to	O
resample	O
cases	O
means	O
that	O
each	O
bootstrap	B-Application
sample	I-Application
will	O
lose	O
some	O
information	O
.	O
</s>
<s>
As	O
such	O
,	O
alternative	O
bootstrap	B-Application
procedures	O
should	O
be	O
considered	O
.	O
</s>
<s>
Bootstrapping	B-Application
can	O
be	O
interpreted	O
in	O
a	O
Bayesian	O
framework	O
using	O
a	O
scheme	O
that	O
creates	O
new	O
data	B-General_Concept
sets	I-General_Concept
through	O
reweighting	O
the	O
initial	O
data	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
data	O
points	O
,	O
the	O
weighting	O
assigned	O
to	O
data	O
point	O
in	O
a	O
new	O
data	B-General_Concept
set	I-General_Concept
is	O
,	O
where	O
is	O
a	O
low-to-high	O
ordered	O
list	O
of	O
uniformly	O
distributed	O
random	O
numbers	O
on	O
,	O
preceded	O
by	O
0	O
and	O
succeeded	O
by	O
1	O
.	O
</s>
<s>
The	O
distributions	O
of	O
a	O
parameter	O
inferred	O
from	O
considering	O
many	O
such	O
data	B-General_Concept
sets	I-General_Concept
are	O
then	O
interpretable	O
as	O
posterior	O
distributions	O
on	O
that	O
parameter	O
.	O
</s>
<s>
This	O
is	O
equivalent	O
to	O
sampling	O
from	O
a	O
kernel	B-General_Concept
density	I-General_Concept
estimate	I-General_Concept
of	O
the	O
data	O
.	O
</s>
<s>
Assume	O
K	O
to	O
be	O
a	O
symmetric	O
kernel	B-General_Concept
density	I-General_Concept
function	O
with	O
unit	O
variance	O
.	O
</s>
<s>
Based	O
on	O
the	O
assumption	O
that	O
the	O
original	O
data	B-General_Concept
set	I-General_Concept
is	O
a	O
realization	O
of	O
a	O
random	O
sample	O
from	O
a	O
distribution	O
of	O
a	O
specific	O
parametric	O
type	O
,	O
in	O
this	O
case	O
a	O
parametric	O
model	O
is	O
fitted	O
by	O
parameter	O
θ	O
,	O
often	O
by	O
maximum	O
likelihood	O
,	O
and	O
samples	O
of	O
random	O
numbers	O
are	O
drawn	O
from	O
this	O
fitted	O
model	O
.	O
</s>
<s>
This	O
sampling	O
process	O
is	O
repeated	O
many	O
times	O
as	O
for	O
other	O
bootstrap	B-General_Concept
methods	I-General_Concept
.	O
</s>
<s>
Considering	O
the	O
centered	O
sample	O
mean	O
in	O
this	O
case	O
,	O
the	O
random	O
sample	O
original	O
distribution	O
function	O
is	O
replaced	O
by	O
a	O
bootstrap	B-Application
random	O
sample	O
with	O
function	O
,	O
and	O
the	O
probability	O
distribution	O
of	O
is	O
approximated	O
by	O
that	O
of	O
,	O
where	O
,	O
which	O
is	O
the	O
expectation	O
corresponding	O
to	O
.	O
</s>
<s>
The	O
use	O
of	O
a	O
parametric	O
model	O
at	O
the	O
sampling	O
stage	O
of	O
the	B-Application
bootstrap	I-Application
methodology	O
leads	O
to	O
procedures	O
which	O
are	O
different	O
from	O
those	O
obtained	O
by	O
applying	O
basic	O
statistical	O
theory	O
to	O
inference	O
for	O
the	O
same	O
model	O
.	O
</s>
<s>
Another	O
approach	O
to	O
bootstrapping	B-Application
in	O
regression	O
problems	O
is	O
to	O
resample	O
residuals	O
.	O
</s>
<s>
When	O
data	O
are	O
temporally	O
correlated	O
,	O
straightforward	O
bootstrapping	B-Application
destroys	O
the	O
inherent	O
correlations	O
.	O
</s>
<s>
The	O
wild	O
bootstrap	B-Application
,	O
proposed	O
originally	O
by	O
Wu	O
(	O
1986	O
)	O
,	O
is	O
suited	O
when	O
the	O
model	O
exhibits	O
heteroskedasticity	B-General_Concept
.	O
</s>
<s>
The	O
idea	O
is	O
,	O
as	O
the	O
residual	O
bootstrap	B-Application
,	O
to	O
leave	O
the	O
regressors	O
at	O
their	O
sample	O
value	O
,	O
but	O
to	O
resample	O
the	O
response	O
variable	O
based	O
on	O
the	O
residuals	O
values	O
.	O
</s>
<s>
The	O
block	O
bootstrap	B-Application
is	O
used	O
when	O
the	O
data	O
,	O
or	O
the	O
errors	O
in	O
a	O
model	O
,	O
are	O
correlated	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
a	O
simple	O
case	O
or	O
residual	O
resampling	B-General_Concept
will	O
fail	O
,	O
as	O
it	O
is	O
not	O
able	O
to	O
replicate	O
the	O
correlation	O
in	O
the	O
data	O
.	O
</s>
<s>
The	O
block	O
bootstrap	B-Application
tries	O
to	O
replicate	O
the	O
correlation	O
by	O
resampling	B-General_Concept
inside	O
blocks	O
of	O
data	O
(	O
see	O
Blocking	O
(	O
statistics	O
)	O
)	O
.	O
</s>
<s>
The	O
block	O
bootstrap	B-Application
has	O
been	O
used	O
mainly	O
with	O
data	O
correlated	O
in	O
time	O
(	O
i.e.	O
</s>
<s>
In	O
the	O
(	O
simple	O
)	O
block	O
bootstrap	B-Application
,	O
the	O
variable	O
of	O
interest	O
is	O
split	O
into	O
non-overlapping	O
blocks	O
.	O
</s>
<s>
In	O
the	O
moving	O
block	O
bootstrap	B-Application
,	O
introduced	O
by	O
Künsch	O
(	O
1989	O
)	O
,	O
data	O
is	O
split	O
into	O
n−b+1	O
overlapping	O
blocks	O
of	O
length	O
b	O
:	O
Observation	O
1	O
to	O
b	O
will	O
be	O
block	O
1	O
,	O
observation	O
2	O
to	O
b+1	O
will	O
be	O
block	O
2	O
,	O
etc	O
.	O
</s>
<s>
Then	O
aligning	O
these	O
n/b	O
blocks	O
in	O
the	O
order	O
they	O
were	O
picked	O
,	O
will	O
give	O
the	B-Application
bootstrap	I-Application
observations	O
.	O
</s>
<s>
This	O
bootstrap	B-Application
works	O
with	O
dependent	O
data	O
,	O
however	O
,	O
the	O
bootstrapped	O
observations	O
will	O
not	O
be	O
stationary	O
anymore	O
by	O
construction	O
.	O
</s>
<s>
This	O
method	O
is	O
known	O
as	O
the	O
stationary	O
bootstrap	B-Application
.	O
</s>
<s>
Other	O
related	O
modifications	O
of	O
the	O
moving	O
block	O
bootstrap	B-Application
are	O
the	O
Markovian	O
bootstrap	B-Application
and	O
a	O
stationary	O
bootstrap	B-Application
method	I-Application
that	O
matches	O
subsequent	O
blocks	O
based	O
on	O
standard	O
deviation	O
matching	O
.	O
</s>
<s>
Vinod	O
(	O
2006	O
)	O
,	O
presents	O
a	O
method	O
that	O
bootstraps	B-Application
time	O
series	O
data	O
using	O
maximum	O
entropy	O
principles	O
satisfying	O
the	O
Ergodic	O
theorem	O
with	O
mean-preserving	O
and	O
mass-preserving	O
constraints	O
.	O
</s>
<s>
The	O
structure	O
of	O
the	O
block	O
bootstrap	B-Application
is	O
easily	O
obtained	O
(	O
where	O
the	O
block	O
just	O
corresponds	O
to	O
the	O
group	O
)	O
,	O
and	O
usually	O
only	O
the	O
groups	O
are	O
resampled	O
,	O
while	O
the	O
observations	O
within	O
the	O
groups	O
are	O
left	O
unchanged	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
is	O
a	O
powerful	O
technique	O
although	O
may	O
require	O
substantial	O
computing	O
resources	O
in	O
both	O
time	O
and	O
memory	O
.	O
</s>
<s>
They	O
can	O
generally	O
be	O
combined	O
with	O
many	O
of	O
the	O
different	O
types	O
of	O
Bootstrap	B-Application
schemes	O
and	O
various	O
choices	O
of	O
statistics	O
.	O
</s>
<s>
The	O
ordinary	O
bootstrap	B-Application
requires	O
the	O
random	O
selection	O
of	O
n	O
elements	O
from	O
a	O
list	O
,	O
which	O
is	O
equivalent	O
to	O
drawing	O
from	O
a	O
multinomial	O
distribution	O
.	O
</s>
<s>
For	O
large	O
values	O
of	O
n	O
,	O
the	O
Poisson	O
bootstrap	B-Application
is	O
an	O
efficient	O
method	O
of	O
generating	O
bootstrapped	O
data	B-General_Concept
sets	I-General_Concept
.	O
</s>
<s>
When	O
generating	O
a	O
single	O
bootstrap	B-Application
sample	I-Application
,	O
instead	O
of	O
randomly	O
drawing	O
from	O
the	O
sample	O
data	O
with	O
replacement	O
,	O
each	O
data	O
point	O
is	O
assigned	O
a	O
random	O
weight	O
distributed	O
according	O
to	O
the	O
Poisson	O
distribution	O
with	O
.	O
</s>
<s>
This	O
method	O
also	O
lends	O
itself	O
well	O
to	O
streaming	O
data	O
and	O
growing	O
data	B-General_Concept
sets	I-General_Concept
,	O
since	O
the	O
total	O
number	O
of	O
samples	O
does	O
not	O
need	O
to	O
be	O
known	O
in	O
advance	O
of	O
beginning	O
to	O
take	O
bootstrap	B-Application
samples	I-Application
.	O
</s>
<s>
For	O
large	O
enough	O
n	O
,	O
the	O
results	O
are	O
relatively	O
similar	O
to	O
the	O
original	O
bootstrap	B-Application
estimations	O
.	O
</s>
<s>
A	O
way	O
to	O
improve	O
on	O
the	O
poisson	O
bootstrap	B-Application
,	O
termed	O
"	O
sequential	O
bootstrap	B-Application
"	O
,	O
is	O
by	O
taking	O
the	O
first	O
samples	O
so	O
that	O
the	O
proportion	O
of	O
unique	O
values	O
is	O
≈	O
0.632	O
of	O
the	O
original	O
sample	O
size	O
n	O
.	O
This	O
provides	O
a	O
distribution	O
with	O
main	O
empirical	O
characteristics	O
being	O
within	O
a	O
distance	O
of	O
.	O
</s>
<s>
This	O
is	O
related	O
to	O
the	O
reduced	O
bootstrap	B-Application
method	I-Application
.	O
</s>
<s>
For	O
massive	O
data	B-General_Concept
sets	I-General_Concept
,	O
it	O
is	O
often	O
computationally	O
prohibitive	O
to	O
hold	O
all	O
the	O
sample	O
data	O
in	O
memory	O
and	O
resample	O
from	O
the	O
sample	O
data	O
.	O
</s>
<s>
The	O
Bag	O
of	O
Little	O
Bootstraps	B-Application
(	O
BLB	O
)	O
provides	O
a	O
method	O
of	O
pre-aggregating	O
data	O
before	O
bootstrapping	B-Application
to	O
reduce	O
computational	O
constraints	O
.	O
</s>
<s>
This	O
works	O
by	O
partitioning	O
the	O
data	B-General_Concept
set	I-General_Concept
into	O
equal-sized	O
buckets	O
and	O
aggregating	O
the	O
data	O
within	O
each	O
bucket	O
.	O
</s>
<s>
This	O
pre-aggregated	O
data	B-General_Concept
set	I-General_Concept
becomes	O
the	O
new	O
sample	O
data	O
over	O
which	O
to	O
draw	O
samples	O
with	O
replacement	O
.	O
</s>
<s>
This	O
method	O
is	O
similar	O
to	O
the	O
Block	O
Bootstrap	B-Application
,	O
but	O
the	O
motivations	O
and	O
definitions	O
of	O
the	O
blocks	O
are	O
very	O
different	O
.	O
</s>
<s>
Under	O
certain	O
assumptions	O
,	O
the	O
sample	B-General_Concept
distribution	I-General_Concept
should	O
approximate	O
the	O
full	O
bootstrapped	O
scenario	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
distribution	O
of	O
a	O
point	O
estimator	O
of	O
a	O
population	O
parameter	O
has	O
been	O
used	O
to	O
produce	O
a	O
bootstrapped	O
confidence	O
interval	O
for	O
the	O
parameter	O
's	O
true	O
value	O
if	O
the	O
parameter	O
can	O
be	O
written	O
as	O
a	O
function	O
of	O
the	O
population	O
's	O
distribution	O
.	O
</s>
<s>
Popular	O
families	O
of	O
point-estimators	O
include	O
mean-unbiased	O
minimum-variance	O
estimators	O
,	O
median-unbiased	O
estimators	O
,	O
Bayesian	B-General_Concept
estimators	I-General_Concept
(	O
for	O
example	O
,	O
the	O
posterior	O
distribution	O
's	O
mode	O
,	O
median	O
,	O
mean	O
)	O
,	O
and	O
maximum-likelihood	O
estimators	O
.	O
</s>
<s>
Asymptotic	O
theory	O
suggests	O
techniques	O
that	O
often	O
improve	O
the	O
performance	O
of	O
bootstrapped	O
estimators	O
;	O
the	O
bootstrapping	B-Application
of	O
a	O
maximum-likelihood	O
estimator	O
may	O
often	O
be	O
improved	O
using	O
transformations	O
related	O
to	O
pivotal	O
quantities	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
distribution	O
of	O
a	O
parameter-estimator	O
has	O
been	O
used	O
to	O
calculate	O
confidence	O
intervals	O
for	O
its	O
population-parameter	O
.	O
</s>
<s>
Bias	O
:	O
The	B-Application
bootstrap	I-Application
distribution	O
and	O
the	O
sample	O
may	O
disagree	O
systematically	O
,	O
in	O
which	O
case	O
bias	O
may	O
occur	O
.	O
</s>
<s>
If	O
the	B-Application
bootstrap	I-Application
distribution	O
of	O
an	O
estimator	O
is	O
symmetric	O
,	O
then	O
percentile	O
confidence-interval	O
are	O
often	O
used	O
;	O
such	O
intervals	O
are	O
appropriate	O
especially	O
for	O
median-unbiased	O
estimators	O
of	O
minimum	O
risk	O
(	O
with	O
respect	O
to	O
an	O
absolute	O
loss	O
function	O
)	O
.	O
</s>
<s>
Bias	O
in	O
the	B-Application
bootstrap	I-Application
distribution	O
will	O
lead	O
to	O
bias	O
in	O
the	O
confidence	O
interval	O
.	O
</s>
<s>
Otherwise	O
,	O
if	O
the	B-Application
bootstrap	I-Application
distribution	O
is	O
non-symmetric	O
,	O
then	O
percentile	O
confidence	O
intervals	O
are	O
often	O
inappropriate	O
.	O
</s>
<s>
There	O
are	O
several	O
methods	O
for	O
constructing	O
confidence	O
intervals	O
from	O
the	B-Application
bootstrap	I-Application
distribution	O
of	O
a	O
real	O
parameter	O
:	O
</s>
<s>
Basic	O
bootstrap	B-Application
,	O
also	O
known	O
as	O
the	O
Reverse	O
Percentile	O
Interval	O
.	O
</s>
<s>
The	O
basic	O
bootstrap	B-Application
is	O
a	O
simple	O
scheme	O
to	O
construct	O
the	O
confidence	O
interval	O
:	O
one	O
simply	O
takes	O
the	O
empirical	O
quantiles	O
from	O
the	B-Application
bootstrap	I-Application
distribution	O
of	O
the	O
parameter	O
(	O
see	O
Davison	O
and	O
Hinkley	O
1997	O
,	O
equ	O
.	O
</s>
<s>
Percentile	O
bootstrap	B-Application
.	O
</s>
<s>
The	O
percentile	O
bootstrap	B-Application
proceeds	O
in	O
a	O
similar	O
way	O
to	O
the	O
basic	O
bootstrap	B-Application
,	O
using	O
percentiles	O
of	O
the	B-Application
bootstrap	I-Application
distribution	O
,	O
but	O
with	O
a	O
different	O
formula	O
(	O
note	O
the	O
inversion	O
of	O
the	O
left	O
and	O
right	O
quantiles	O
)	O
:	O
</s>
<s>
It	O
will	O
work	O
well	O
in	O
cases	O
where	O
the	B-Application
bootstrap	I-Application
distribution	O
is	O
symmetrical	O
and	O
centered	O
on	O
the	O
observed	O
statistic	O
and	O
where	O
the	O
sample	O
statistic	O
is	O
median-unbiased	O
and	O
has	O
maximum	O
concentration	O
(	O
or	O
minimum	O
risk	O
with	O
respect	O
to	O
an	O
absolute	O
value	O
loss	O
function	O
)	O
.	O
</s>
<s>
Studentized	O
bootstrap	B-Application
.	O
</s>
<s>
The	O
studentized	O
bootstrap	B-Application
,	O
also	O
called	O
bootstrap-t	O
,	O
is	O
computed	O
analogously	O
to	O
the	O
standard	O
confidence	O
interval	O
,	O
but	O
replaces	O
the	O
quantiles	O
from	O
the	O
normal	O
or	O
student	O
approximation	O
by	O
the	O
quantiles	O
from	O
the	B-Application
bootstrap	I-Application
distribution	O
of	O
the	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
(	O
see	O
Davison	O
and	O
Hinkley	O
1997	O
,	O
equ	O
.	O
</s>
<s>
where	O
denotes	O
the	O
percentile	O
of	O
the	O
bootstrapped	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
,	O
and	O
is	O
the	O
estimated	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
coefficient	O
in	O
the	O
original	O
model	O
.	O
</s>
<s>
it	O
does	O
not	O
depend	O
on	O
nuisance	O
parameters	O
as	O
the	O
t-test	B-General_Concept
follows	O
asymptotically	O
a	O
N(0,1 )	O
distribution	O
)	O
,	O
unlike	O
the	O
percentile	O
bootstrap	B-Application
.	O
</s>
<s>
Bias-corrected	O
bootstrap	B-Application
–	O
adjusts	O
for	O
bias	O
in	O
the	B-Application
bootstrap	I-Application
distribution	O
.	O
</s>
<s>
Accelerated	O
bootstrap	B-Application
–	O
The	O
bias-corrected	O
and	O
accelerated	O
(	O
BCa	O
)	O
bootstrap	B-Application
,	O
by	O
Efron	O
(	O
1987	O
)	O
,	O
adjusts	O
for	O
both	O
bias	O
and	O
skewness	B-General_Concept
in	O
the	B-Application
bootstrap	I-Application
distribution	O
.	O
</s>
<s>
Create	O
two	O
new	O
data	B-General_Concept
sets	I-General_Concept
whose	O
values	O
are	O
and	O
where	O
is	O
the	O
mean	O
of	O
the	O
combined	O
sample	O
.	O
</s>
<s>
The	O
data	B-General_Concept
set	I-General_Concept
contains	O
two	O
outliers	O
,	O
which	O
greatly	O
influence	O
the	O
sample	O
mean	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
distribution	O
for	O
Newcomb	O
's	O
data	O
appears	O
below	O
.	O
</s>
<s>
We	O
can	O
reduce	O
the	O
discreteness	O
of	O
the	B-Application
bootstrap	I-Application
distribution	O
by	O
adding	O
a	O
small	O
amount	O
of	O
random	O
noise	O
to	O
each	O
bootstrap	B-Application
sample	I-Application
.	O
</s>
<s>
This	O
means	O
that	O
samples	O
taken	O
from	O
the	B-Application
bootstrap	I-Application
distribution	O
will	O
have	O
a	O
variance	O
which	O
is	O
,	O
on	O
average	O
,	O
equal	O
to	O
the	O
variance	O
of	O
the	O
total	O
population	O
.	O
</s>
<s>
Histograms	O
of	O
the	B-Application
bootstrap	I-Application
distribution	O
and	O
the	O
smooth	O
bootstrap	B-Application
distribution	O
appear	O
below	O
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
distribution	O
of	O
the	O
sample-median	O
has	O
only	O
a	O
small	O
number	O
of	O
values	O
.	O
</s>
<s>
The	O
smoothed	O
bootstrap	B-Application
distribution	O
has	O
a	O
richer	O
support	O
.	O
</s>
<s>
However	O
,	O
note	O
that	O
whether	O
the	O
smoothed	O
or	O
standard	O
bootstrap	B-Application
procedure	O
is	O
favorable	O
is	O
case-by-case	O
and	O
is	O
shown	O
to	O
depend	O
on	O
both	O
the	O
underlying	O
distribution	O
function	O
and	O
on	O
the	O
quantity	O
being	O
estimated	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
the	O
bootstrapped	O
95%	O
(	O
percentile	O
)	O
confidence-interval	O
for	O
the	O
population	O
median	O
is	O
(	O
26	O
,	O
28.5	O
)	O
,	O
which	O
is	O
close	O
to	O
the	O
interval	O
for	O
(	O
25.98	O
,	O
28.46	O
)	O
for	O
the	O
smoothed	O
bootstrap	B-Application
.	O
</s>
<s>
The	B-Application
bootstrap	I-Application
is	O
distinguished	O
from	O
:	O
</s>
<s>
cross-validation	B-Application
,	O
in	O
which	O
the	O
parameters	O
(	O
e.g.	O
,	O
regression	O
weights	O
,	O
factor	O
loadings	O
)	O
that	O
are	O
estimated	O
in	O
one	O
subsample	O
are	O
applied	O
to	O
another	O
subsample	O
.	O
</s>
<s>
For	O
more	O
details	O
see	O
resampling	B-General_Concept
.	O
</s>
<s>
Bootstrap	B-Algorithm
aggregating	I-Algorithm
(	O
bagging	B-Algorithm
)	O
is	O
a	O
meta-algorithm	B-Algorithm
based	O
on	O
averaging	O
model	O
predictions	O
obtained	O
from	O
models	O
trained	O
on	O
multiple	O
bootstrap	B-Application
samples	I-Application
.	O
</s>
<s>
Given	O
an	O
r-sample	O
statistic	O
,	O
one	O
can	O
create	O
an	O
n-sample	O
statistic	O
by	O
something	O
similar	O
to	O
bootstrapping	B-Application
(	O
taking	O
the	O
average	O
of	O
the	O
statistic	O
over	O
all	O
subsamples	O
of	O
size	O
r	O
)	O
.	O
</s>
